Two way analysis of binary data (repeated) I have a data set that is measuring a yes(1)/no(0) outcome across six time points and under two conditions. We have 15 participants that completed all time points under the two conditions.
Disclaimer; I have been given this data set to help analyse (part of some pilot testing), and am fully aware the sample size is low, but am interested in this line of research going forward into larger testing.
I think we are effectively looking at a two-way repeated measures analysis (in ANOVA terminology), but have binary data.
Other than re-design the study, are there any suggestions about whether there is a relevant statistical test? 
Thank you
 A: For the set up you described, you have two model options: 


*

*Mixed effects binary logistic regression model;

*GEE binary logistic regression model.


See https://www.cscu.cornell.edu/news/statnews/stnews76.pdf for a brief discussion of the two types of models.
The set up is not completely clear, but everything suggested in my answer would work if the six time points are the only ones you are interested in (rather than a subset of all time points of interest).
Both models would include main effects for condition and time, as well as an interaction between condition and time.  
Depending on your research hypotheses, you could set up contrasts of interests based on the model coefficients in order to test these hypotheses based on the postulated model. 
The models would be fitted using different procedures and their coefficients would have different interpretations.  Both models take into account the correlation of the outcome values  coming from the same subject.
For both models, the coefficients could be exponentiated to obtain odds ratios - however, the mixed effects model will produce subject-specific odds ratios while the GEE model will produce population-average odds ratios. (We're talking about the odds of an Yes rather than a No answer - these odds could be compared between conditions at each time point, for example.) 
Which model you decide to use will depend on the scope of your inferences.  The article Subject specific and population average models for binary longitudinal data: a tutorial by Camille Szmaragd, Paul Clarke and Fiona Steele provides some guidance on when you might want to use each type of model: http://journals.sfu.ca/llcs/index.php/llcs/article/viewFile/249/238.
